If A and B are two events such that P(A) =0.2 , P(B) = 0.4 and `P( A cup B) = 0.6` then P(A/B) is equal to

Options

A. 0
B. `0.5`
C. `0.3`
D. `0.8`

Answer

To solve the problem, we need to find \( P(A|B) \), which is the conditional probability of event A given event B. The formula for conditional probability is:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

Where:
- \( P(A \cap B) \) is the probability of both A and B occurring.
- \( P(B) \) is the probability of event B.

### Step 1: Identify the given probabilities
We are given:
- \( P(A) = 0.2 \)
- \( P(B) = 0.4 \)
- \( P(A \cup B) = 0.6 \)

### Step 2: Use the formula for the union of two events
We know the relationship between the probabilities of the union of two events and their intersection:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

### Step 3: Substitute the known values into the equation
Substituting the known values into the equation:

\[ 0.6 = 0.2 + 0.4 - P(A \cap B) \]

### Step 4: Simplify to find \( P(A \cap B) \)
Now, simplify the equation:

\[ 0.6 = 0.6 - P(A \cap B) \]

Rearranging gives:

\[ P(A \cap B) = 0.2 + 0.4 - 0.6 \]
\[ P(A \cap B) = 0.0 \]

### Step 5: Calculate \( P(A|B) \)
Now we can substitute \( P(A \cap B) \) and \( P(B) \) into the conditional probability formula:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.0}{0.4} \]

### Step 6: Final result
Thus, we find:

\[ P(A|B) = 0 \]

### Summary
The probability \( P(A|B) \) is equal to 0.

---